I finally got it fixed by following the instructions in this post: https://productforums.google.com/forum/#!category-topic/gmail/Hykf1d7KxaM
It was indeed related to 2-step verification, as I had guessed some time ago. -- You received this message because you are subscribed to the Google Groups "Gmail-Users" group. To unsubscribe from this group and stop receiving emails from it, send an email to gmail-users+unsubscr...@googlegroups.com. To post to this group, send email to gmail-users@googlegroups.com. Visit this group at http://groups.google.com/group/gmail-users. For more options, visit https://groups.google.com/groups/opt_out.
